Collaborative filtering

A distinction is often made between two forms of data collection for recommendation systems. Explicit feedback relies on the user giving explicit signals about their preferences i.e. review ratings. Where as, implicit feedback refers to non-explicit signals of preference e.g. user watch-time. Traditionally, recommender systems can be split into three types:

  • Collaborative filtering (CF): CF produces recommendations based on the knowledge of users’ attitudes towards items, that is, it uses the “wisdom of the crowd” to recommend items.

  • Content-based (CB): CB recommender systems focus on the attributes of the items to recommend other items similar to what the user likes, based on their previous actions or explicit feedback.

  • Hybrid recommendation systems: Hybrid methods are a combination of CB recommending and CF methods

In many applications, content-based features are not easy to extract, and thus, collaborative filtering approaches are preferred. Thus, we will only explore collaborative filtering methods from now on.

CF methods typically fall into three types, memory-based, model-based and more recently deep-learning based (Su & Khoshgoftaar, 2009, He et al., 2017). Neighbour-based CF and item-based/user-based top-N recommendations are typical examples of memory-based systems that utilises user rating data to compute the similarity between users or items. As mentioned previously, common model-based approaches include Bayesian networks, latent semantic models and markov decision processes. In this investigation, we will utilise a weighted matrix factorization approach. Later on, we will generalize the matrix factorization algorithm via a non-linear neural architecture (a softmax model).

However, there are a number of limitations to our approaches such as the inability to model the order of interactions. For instance, Markov chain algorithms (Rendle et al., 2010) can not only encode the same information as traditional CF methods but also the order in which user’s interacted with the items. Furthermore, the sparsity of the frequency matrix (described later on), makes computations prohibitly expensive in real-world settings, without some optimization.

Setup

The next few code cells details the initial preparatory steps needed for the development of our collaborative filtering models, namely importing the required libraries; scaling the ids of users and artists;constructing a indicator variable for presence of user-artist interaction;finding the most assigned tag of an artist.

from __future__ import print_function
import numpy as np
import pandas as pd
import collections
from IPython import display
from matplotlib import pyplot as plt
import sklearn
import sklearn.manifold
import tensorflow.compat.v1 as tf
tf.disable_v2_behavior()
tf.logging.set_verbosity(tf.logging.ERROR)

# Add some convenience functions to Pandas DataFrame.
pd.options.display.max_rows = 10
pd.options.display.float_format = '{:.3f}'.format

# Install Altair and activate its colab renderer.
print("Installing Altair...")
!pip install git+git://github.com/altair-viz/altair.git
import altair as alt
alt.data_transformers.enable('default', max_rows=None)
alt.renderers.enable('colab')
print("Done installing Altair.")
2021-11-28 16:50:11.340001: W tensorflow/stream_executor/platform/default/dso_loader.cc:64] Could not load dynamic library 'libcudart.so.11.0'; dlerror: libcudart.so.11.0: cannot open shared object file: No such file or directory
2021-11-28 16:50:11.340047: I tensorflow/stream_executor/cuda/cudart_stub.cc:29] Ignore above cudart dlerror if you do not have a GPU set up on your machine.
WARNING:tensorflow:From /opt/hostedtoolcache/Python/3.7.12/x64/lib/python3.7/site-packages/tensorflow/python/compat/v2_compat.py:111: disable_resource_variables (from tensorflow.python.ops.variable_scope) is deprecated and will be removed in a future version.
Instructions for updating:
non-resource variables are not supported in the long term
Installing Altair...
Collecting git+git://github.com/altair-viz/altair.git
  Cloning git://github.com/altair-viz/altair.git to /tmp/pip-req-build-9au_0mgx
  Running command git clone --filter=blob:none -q git://github.com/altair-viz/altair.git /tmp/pip-req-build-9au_0mgx
  Resolved git://github.com/altair-viz/altair.git to commit a987d04e276106f62d4247ea48a1fcead2d06636
  Installing build dependencies ... ?25l-
 \
 |
 /
 done
?25h  Getting requirements to build wheel ... ?25l-
 done
?25h  Preparing metadata (pyproject.toml) ... ?25l-
 done
?25hRequirement already satisfied: jsonschema<4.0,>=3.0 in /opt/hostedtoolcache/Python/3.7.12/x64/lib/python3.7/site-packages (from altair==4.2.0.dev0) (3.2.0)
Requirement already satisfied: pandas>=0.18 in /opt/hostedtoolcache/Python/3.7.12/x64/lib/python3.7/site-packages (from altair==4.2.0.dev0) (1.3.4)
Requirement already satisfied: entrypoints in /opt/hostedtoolcache/Python/3.7.12/x64/lib/python3.7/site-packages (from altair==4.2.0.dev0) (0.3)
Collecting toolz
  Downloading toolz-0.11.2-py3-none-any.whl (55 kB)
?25l

     |█████▉                          | 10 kB 29.6 MB/s eta 0:00:01
     |███████████▊                    | 20 kB 23.6 MB/s eta 0:00:01
     |█████████████████▋              | 30 kB 17.4 MB/s eta 0:00:01
     |███████████████████████▌        | 40 kB 7.0 MB/s eta 0:00:01 
     |█████████████████████████████▍  | 51 kB 8.2 MB/s eta 0:00:01 
     |████████████████████████████████| 55 kB 5.7 MB/s             
?25hRequirement already satisfied: numpy in /opt/hostedtoolcache/Python/3.7.12/x64/lib/python3.7/site-packages (from altair==4.2.0.dev0) (1.21.4)
Requirement already satisfied: jinja2 in /opt/hostedtoolcache/Python/3.7.12/x64/lib/python3.7/site-packages (from altair==4.2.0.dev0) (3.0.3)
Requirement already satisfied: pyrsistent>=0.14.0 in /opt/hostedtoolcache/Python/3.7.12/x64/lib/python3.7/site-packages (from jsonschema<4.0,>=3.0->altair==4.2.0.dev0) (0.18.0)
Requirement already satisfied: setuptools in /opt/hostedtoolcache/Python/3.7.12/x64/lib/python3.7/site-packages (from jsonschema<4.0,>=3.0->altair==4.2.0.dev0) (47.1.0)
Requirement already satisfied: importlib-metadata in /opt/hostedtoolcache/Python/3.7.12/x64/lib/python3.7/site-packages (from jsonschema<4.0,>=3.0->altair==4.2.0.dev0) (4.8.2)
Requirement already satisfied: six>=1.11.0 in /opt/hostedtoolcache/Python/3.7.12/x64/lib/python3.7/site-packages (from jsonschema<4.0,>=3.0->altair==4.2.0.dev0) (1.16.0)
Requirement already satisfied: attrs>=17.4.0 in /opt/hostedtoolcache/Python/3.7.12/x64/lib/python3.7/site-packages (from jsonschema<4.0,>=3.0->altair==4.2.0.dev0) (21.2.0)
Requirement already satisfied: pytz>=2017.3 in /opt/hostedtoolcache/Python/3.7.12/x64/lib/python3.7/site-packages (from pandas>=0.18->altair==4.2.0.dev0) (2021.3)
Requirement already satisfied: python-dateutil>=2.7.3 in /opt/hostedtoolcache/Python/3.7.12/x64/lib/python3.7/site-packages (from pandas>=0.18->altair==4.2.0.dev0) (2.8.2)
Requirement already satisfied: MarkupSafe>=2.0 in /opt/hostedtoolcache/Python/3.7.12/x64/lib/python3.7/site-packages (from jinja2->altair==4.2.0.dev0) (2.0.1)
Requirement already satisfied: zipp>=0.5 in /opt/hostedtoolcache/Python/3.7.12/x64/lib/python3.7/site-packages (from importlib-metadata->jsonschema<4.0,>=3.0->altair==4.2.0.dev0) (3.6.0)
Requirement already satisfied: typing-extensions>=3.6.4 in /opt/hostedtoolcache/Python/3.7.12/x64/lib/python3.7/site-packages (from importlib-metadata->jsonschema<4.0,>=3.0->altair==4.2.0.dev0) (4.0.0)
Building wheels for collected packages: altair
  Building wheel for altair (pyproject.toml) ... ?25l-
 \
 |
 done
?25h  Created wheel for altair: filename=altair-4.2.0.dev0-py3-none-any.whl size=812168 sha256=aba552135028ce108d5aecb0265efd4c9128b68ad5a47bbaeb9740ae142fcda7
  Stored in directory: /tmp/pip-ephem-wheel-cache-j3ilv0e3/wheels/06/13/e0/5bd72c969fe3954ee1561739e5c58e2ddfe5c10fcdffb12faa
Successfully built altair
Installing collected packages: toolz, altair
Successfully installed altair-4.2.0.dev0 toolz-0.11.2
Done installing Altair.
# NEEDED FOR GOOGLE COLAB
# from google.colab import auth
#from google.colab import drive
# import gspread
# from oauth2client.client import GoogleCredentials

# drive.mount('/content/drive/')
# os.chdir("/content/drive/My Drive/DCU/fouth_year/advanced_machine_learning/music-recommodation-system")

Helper functions

def calculate_sparsity(M):
    """
    Computes sparsity of frequency matrix
    """
    matrix_size = len((M['userID'].unique())) * len((M['artistID'].unique()))  # Number of possible interactions in the matrix
    num_plays = len(M['weight']) # Number of weights
    sparsity = (float(num_plays/matrix_size))
    return sparsity
def build_music_sparse_tensor(music_df):
  """
  Args:
    ratings_df: a pd.DataFrame with `userID`, `artistID` and `weight` columns.
    num_rows: an integer representing the number of rows in the frequency matrix
    num_rows: an integer representing the number of columns in the frequency matrix
  Returns:
    a tf.SparseTensor representing the feedback matrix.
  """
  indices = music_df[['userID', 'artistID']].values
  values = music_df['weight'].values
  return tf.SparseTensor(
      indices=indices,
      values=values,
      dense_shape=[num_users, num_artist])
def preproces_ids(music_df):
  """
  Args:
    ratings_df: a pd.DataFrame with `userID`, `artistID` and `weight` columns.
  Returns:
    a pd.DataFrame where userIDs and artistIDs now start at 1 
      and end at n and m (defined above), respectively
    two dictionary preserving the orginal ids. 
  """
  unique_user_ids_list = sorted(music_df['userID'].unique())
  print(unique_user_ids_list[0])

  unique_user_ids = dict(zip(range(0, len(unique_user_ids_list) ),unique_user_ids_list))
  unique_user_ids_switched = dict(zip(unique_user_ids_list, range(0, len(unique_user_ids) )))
  
  unique_artist_ids_list = sorted(music_df['artistID'].unique())
  unique_artist_ids = dict(zip(range(0, len(unique_artist_ids_list) ),unique_artist_ids_list))
  unique_artist_ids_switched = dict(zip(unique_artist_ids_list, range(0, len(unique_artist_ids_list) )))

  music_df['userID'] = music_df['userID'].map(unique_user_ids_switched)
  music_df['artistID'] = music_df['artistID'].map(unique_artist_ids_switched)

  return music_df, unique_user_ids, unique_artist_ids
def split_dataframe(df, holdout_fraction=0.1):
  """Splits a DataFrame into training and test sets.
  Args:
    df: a dataframe.
    holdout_fraction: fraction of dataframe rows to use in the test set.
  Returns:
    train: dataframe for training
    test: dataframe for testing
  """
  test = df.sample(frac=holdout_fraction, replace=False)
  train = df[~df.index.isin(test.index)]
  return train, test

Traditional recommender system development relies on explicit feedback. Many models were designed to tackle this issue as a regression problem. For instance, the input of the model would be a matrix \(F_{nm}\) denoting user’s (m) preference of items (n) on a scale. In the classic movie ratings example, this preference would be users giving a 1-to-5 star rating to different movies.

This dataset contains implicit feedback: that is, observed logs of user interactions with items, in this instance user’s listening counts to artists. However, implicit feedback does not signal negativity, in the same way as a 1-star rating would. In our data, a user could listen to song of an artist a limited number of times. But that does not necessarily mean that the particular user has an aversion to that artist i.e. it could be part of a curated playlist by another user. Therefore, we decide to construct a binary matrix, which has a value of one if the observation is observed (i.e. a listening count has been logged between an artist and a user). Note, a 0 is not used to describe unobserved artist-user interactions. This is for optimization reasons, explained below.

user_artists = pd.read_csv('data/user_artists.dat', sep='\t')
user_artists['weight'] = 1
artists = pd.read_csv('data/artists.dat', sep='\t')
artists.rename({'id':'artistID'}, inplace=True, axis=1)
user_taggedartists = pd.read_csv(r'data/user_taggedartists-timestamps.dat', sep='\t')
user_taggedartists_years = pd.read_csv(r'data/user_taggedartists.dat', sep='\t')
tags = pd.read_csv(open('data/tags.dat', errors='replace'), sep='\t')
user_taggedartists = pd.merge(user_taggedartists, tags, on=['tagID'])
num_users = user_artists.userID.nunique()
num_artist = artists.artistID.nunique()
collab_filter_df = user_artists

Here, we calculate the top 10 tags by popularity. Then, we assign it to a artist, if the artist has a top 10 tag. If an artist’s tags are not in the top 10, we input ‘N/A’. Note, the next cell can take several mintues to compute.

top_10_tags = user_taggedartists['tagValue'].value_counts().index[0:10]
user_taggedartists['top10TagValue'] = None
for index, row in user_taggedartists.iterrows():
  if row['tagValue'] in top_10_tags:
    user_taggedartists.iloc[index, -1] = row['tagValue']
user_taggedartists.fillna('N/A',inplace=True)
artists = pd.merge(user_taggedartists, artists, on=['artistID'], how='right')[['artistID','name','top10TagValue','tagValue']].fillna('N/A')
artists.groupby(['artistID','name','top10TagValue']).agg(lambda x:x.value_counts().index[0]).reset_index()
artists = artists.drop_duplicates(subset=['artistID'])
assert artists.artistID.nunique() == num_artist
artists.rename({'tagValue':'mostCommonGenre'},axis=1, inplace=True)

We require two matrices or embeddings to compute a similarity measure (one for quires and one for items), but how do we get these two embeddings?

Matrix Factorisation

Figure 2: Data flow chart

First, we need to contsruct the feedback matrix \(F \in R^{m \times n}\), where \(m\) is the number of users and \(n\) is the number of artists. The goal is to two generate two lower-dimensional matrices \(U_{mp}\) and \(V_{np}\) ( with \(p << m\) and \(p << n\)), representing latent user and artist components, so that: $\( F \approx UV^\top \)$

First,we attempt to build the frequency matrix for both training and testing data. tf.SparseTensor is used for efficient representation. Three separate arguments are used to represent a tensor, namely indices, values, dense_shape, where a value \(A_{ij} = a\) is encoded by setting indices[k] = [i, j] and values[k] = a. The last tensor dense_shape is used to specify the shape of the full underlying matrix. Note, as the indices arguments represent row and columns indices, some pre-processing needs to be performed on artist and user IDs. The IDs should start from 0 and end at \(m-1\) and \(n-1\) for users and artists respectively. Presently, userIDs start at 2. Two dictionaries, orginal_artist_ids, orginal_user_ids will preserve the original ids for analysis purposes later on. Assertions and print statements are used to ensure the validity of the transformations.

colab_filter_df, orginal_user_ids, orginal_artist_ids =  preproces_ids(collab_filter_df)
2
colab_filter_df.describe()
userID artistID weight
count 92834.000 92834.000 92834.000
mean 944.222 3235.737 1.000
std 546.751 4197.217 0.000
min 0.000 0.000 1.000
25% 470.000 430.000 1.000
50% 944.000 1237.000 1.000
75% 1416.000 4266.000 1.000
max 1891.000 17631.000 1.000

Next, we caulcate the number of unique artists, userids and sparisty of our proposed frequency matrix, before splitting into training and test subsets. Quite a sparse matrix indeed!

print(f'Number of unqiue users are: {collab_filter_df["userID"].nunique()}')
print(f'Number of unqiue artists are: {collab_filter_df["artistID"].nunique()}')
print(f'Sparsity of our frequency matrix: {calculate_sparsity(collab_filter_df)}')
Number of unqiue users are: 1892
Number of unqiue artists are: 17632
Sparsity of our frequency matrix: 0.002782815119924182
collab_filter_df.to_csv('data/test_user_artists.csv',index=False)
frequency_m_train, frequency_m_test = split_dataframe(colab_filter_df)
frequency_m_train_tensor  = build_music_sparse_tensor(frequency_m_train)
frequency_m_test_tensor  = build_music_sparse_tensor(frequency_m_test)
assert num_users  == frequency_m_train_tensor.shape.as_list()[0] 
assert num_artist == frequency_m_train_tensor.shape.as_list()[1] 
assert num_users == frequency_m_test_tensor.shape.as_list()[0] 
assert num_artist == frequency_m_test_tensor.shape.as_list()[1] 

Training a Matrix factorization model

Per the definition above, \(UV^\top\) approximates \(F\). The Mean Squared Error is used to measure this approximation error. In the notation below, k is used to represent the set of observed listening counts, and K is the number of observed listening counts.

\[ \begin{align*} \text{MSE}(F, UV^\top) = \frac{1}{K}\sum_{(i, j) \in k}{( F_{ij} - (UV^\top)_{ij})^2} \end{align*} \]

However, rather than computing the full prediction matrix, \(UV^\top\) and gathering the entries in the embeddings (corresponding to the observed listening counts) , we only gather the embeddings of the observers pairs and compute their dot products. Thereby, we reduce the complexity from \(O(NM)\) to \(O(Kp)\) where \(p\) is the embedding dimension. Stochastic gradient descent (SGD) is used to minimize the loss (objective) function. The SDG algorithim cycles through the observed listening binary and caulates the prediction according to the following equation.

\[ e_{ui} = F_{ui} - U_{i}V_{j} \]

Then it updates the user and artist as embeddings as shown in the following equations.

\[ U_{i} \leftarrow U_{i} + \alpha (e_{ui}V_{j} - \beta U_{i}) \]
\[ V_{j} \leftarrow V_{j} + \alpha (e_{ui}U_{j} - \beta V_{j}) \]

where \(\alpha\) denotes the learning rate. The algorithim continues untill convergence is found.

Other matrix factorization algorithms functions are also commonly used such as Alternating Least Squares (Takács and Tikk, 2012). A modified version of the aforementioned function known as Weighted Alternating Least Squares (WALS) is slower than SDG but can be parallelised. For the purposes of this investigation, we are not particularly concerned with training times/latency requirements so we proceed with SDG.

We also decide to add regularization to our model, to avoid overfitting. Overfitting occurs when the model tries to fit the training dataset to hard and does not generalize well to unseen or future data. In the context of artist recommendation, fitting the observed listening counts often emphasizes learning high similarity (between artists with many listeners), but a good embedding representation also requires learning low similarity (between artists with few listeners).

First, we define the two classes train_matrix_norm and build_matrix_norm class. The build_matrix_norm class computes the necessary pre-processing steps before we train the model such as specifying the loss metric to optimise and the loss components( e.g. gravity loss for the regularized model) and the initial artist and user embeddings. train_matrix_norm simply trains the models and outputs figures detailing the the loss metrics and components. The methods build_vanilla() and build_reg_model() computes the necessary pre-processing steps for the non-regularized and regularized model.

### Training a Matrix Factorization model
class train_matrix_norm(object):
  """Simple class that represents a matrix normalisation model"""
  def __init__(self, embedding_vars, loss, metrics=None):
    """Initializes a Matrix normalisation model 
    Args:
      embedding_vars: A dictionary of tf.Variables.
      loss: A float Tensor. The loss to optimize.
      metrics: optional list of dictionaries of Tensors. The metrics in each
        dictionary will be plotted in a separate figure during training.
    """
    self._embedding_vars = embedding_vars
    self._loss = loss
    self._metrics = metrics
    self._embeddings = {k: None for k in embedding_vars}
    self._session = None


  @property
  def embeddings(self):
    """The embeddings dictionary."""
    return self._embeddings


  def train(self, num_iterations=100, learning_rate=1.0, plot_results=True,
            optimizer=tf.train.GradientDescentOptimizer):
    """Trains the model.
    Args:
      iterations: number of iterations to run.
      learning_rate: optimizer learning rate.
      plot_results: whether to plot the results at the end of training.
      optimizer: the optimizer to use. Default to SDG
    Returns:
      The metrics dictionary evaluated at the last iteration.
    """
    with self._loss.graph.as_default():
      opt = optimizer(learning_rate)
      train_op = opt.minimize(self._loss)
      local_init_op = tf.group(
          tf.variables_initializer(opt.variables()),
          tf.local_variables_initializer())
      if self._session is None:
        self._session = tf.Session()
        with self._session.as_default():
          self._session.run(tf.global_variables_initializer())
          self._session.run(tf.tables_initializer())
          tf.train.start_queue_runners()

    with self._session.as_default():
      local_init_op.run()
      iterations = []
      metrics = self._metrics or ({},)
      metrics_vals = [collections.defaultdict(list) for _ in self._metrics]

      # Train and append results.
      for i in range(num_iterations + 1):
        _, results = self._session.run((train_op, metrics))
        if (i % 10 == 0) or i == num_iterations:
          print("\r iteration %d: " % i + ", ".join(
                ["%s=%f" % (k, v) for r in results for k, v in r.items()]),
                end='')
          iterations.append(i)
          for metric_val, result in zip(metrics_vals, results):
            for k, v in result.items():
              metric_val[k].append(v)

      for k, v in self._embedding_vars.items():
        self._embeddings[k] = v.eval()

      if plot_results:
        # Plot the metrics.
        num_subplots = len(metrics)+1
        fig = plt.figure()
        fig.set_size_inches(num_subplots*10, 8)
        for i, metric_vals in enumerate(metrics_vals):
          ax = fig.add_subplot(1, num_subplots, i+1)
          for k, v in metric_vals.items():
            ax.plot(iterations, v, label=k)
          ax.set_xlim([1, num_iterations])
          ax.legend()
      return results

class build_matrix_norm():
  """Simple class that represents a matrix normalisation model"""
  def __init__(self, listens, embedding_dim=3, regularization_coeff=.1, gravity_coeff=1.,
    init_stddev=0.1):
    """Initializes a Matrix normalisation model 
    Args:
      listens: the DataFrame of artist listening counts.
      embedding_dim: The dimension of the embedding space.
      regularization_coeff: The regularization coefficient lambda.
      gravity_coeff: The gravity regularization coefficient lambda_g.
    Returns:
      A train_matrix_norm object that uses a regularized loss.
  """
    self._embedding_vars = embedding_vars
    self._loss = loss
    self._metrics = metrics
    self._embeddings = {k: None for k in embedding_vars}
    self._session = None

  def sparse_mean_square_error(sparse_listens, user_embeddings, artist_embeddings):
    """
    Args:
      sparse_listens: A SparseTensor rating matrix, of dense_shape [N, M]
      user_embeddings: A dense Tensor U of shape [N, k] where k is the embedding
        dimension, such that U_i is the embedding of user i.
      artist_embeddings: A dense Tensor V of shape [M, k] where k is the embedding
        dimension, such that V_j is the embedding of movie j.
    Returns:
      A scalar Tensor representing the MSE between the true ratings and the
        model's predictions.
    """
    predictions = tf.gather_nd(
        tf.matmul(user_embeddings, artist_embeddings, transpose_b=True),
        sparse_listens.indices)
    loss = tf.losses.mean_squared_error(sparse_listens.values, predictions)
    return loss
    
  def gravity(U, V):
    """Creates a gravity loss given two embedding matrices."""
    return 1. / (U.shape[0].value*V.shape[0].value) * tf.reduce_sum(
        tf.matmul(U, U, transpose_a=True) * tf.matmul(V, V, transpose_a=True))
  
  def build_vanilla(embedding_dim=3, init_stddev=1.):
    """performs the necessary preprocessing steps for the regularized model.  """ 
    # Initialize the embeddings using a normal distribution.
    U = tf.Variable(tf.random.normal(
      [frequency_m_train_tensor.dense_shape[0], embedding_dim], stddev=init_stddev))
    V = tf.Variable(tf.random.normal(
      [frequency_m_train_tensor.dense_shape[1], embedding_dim], stddev=init_stddev))
    
    embeddings = {"userID": U, "artistID": V}
    error_train = build_matrix_norm.sparse_mean_square_error(frequency_m_train_tensor, U, V)
    error_test = build_matrix_norm.sparse_mean_square_error(frequency_m_test_tensor, U, V)
    metrics = {
        'train_error': error_train,
        'test_error': error_test
    }
    return train_matrix_norm(embeddings, error_train, [metrics])


  def build_reg_model(embedding_dim=3, regularization_coeff=.1, gravity_coeff=1.,
  init_stddev=0.1
  ):
    """performs the necessary preprocessing steps for the regularized model.  """ 
    U = tf.Variable(tf.random.normal(
      [frequency_m_train_tensor.dense_shape[0], embedding_dim], stddev=init_stddev))
    V = tf.Variable(tf.random.normal(
      [frequency_m_train_tensor.dense_shape[1], embedding_dim], stddev=init_stddev))
  
    embeddings = {"userID": U, "artistID": V}

    error_train = build_matrix_norm.sparse_mean_square_error(frequency_m_train_tensor, U, V)
    error_test = build_matrix_norm.sparse_mean_square_error(frequency_m_test_tensor, U, V)
    gravity_loss = gravity_coeff * build_matrix_norm.gravity(U, V)
    regularization_loss = regularization_coeff * (
      tf.reduce_sum(U*U)/U.shape[0].value + tf.reduce_sum(V*V)/V.shape[0].value)
    total_loss = error_train + regularization_loss + gravity_loss
    losses = {
      'train_error_observed': error_train,
      'test_error_observed': error_test,
    }
    loss_components = {
      'observed_loss': error_train,
      'regularization_loss': regularization_loss,
      'gravity_loss': gravity_loss,
    }
    #embeddings = {"userID": U, "artistID": V}

    return train_matrix_norm(embeddings, total_loss, [losses, loss_components])

Vanilla Model (non-regularized)

vanilla_model = build_matrix_norm.build_vanilla(embedding_dim=35,init_stddev=.05)
vanilla_model.train(num_iterations=2000, learning_rate=20.)
2021-11-28 16:52:15.861606: W tensorflow/stream_executor/platform/default/dso_loader.cc:64] Could not load dynamic library 'libcuda.so.1'; dlerror: libcuda.so.1: cannot open shared object file: No such file or directory
2021-11-28 16:52:15.861646: W tensorflow/stream_executor/cuda/cuda_driver.cc:269] failed call to cuInit: UNKNOWN ERROR (303)
2021-11-28 16:52:15.861673: I tensorflow/stream_executor/cuda/cuda_diagnostics.cc:156] kernel driver does not appear to be running on this host (fv-az83-233): /proc/driver/nvidia/version does not exist
2021-11-28 16:52:15.861942: I tensorflow/core/platform/cpu_feature_guard.cc:151] This TensorFlow binary is optimized with oneAPI Deep Neural Network Library (oneDNN) to use the following CPU instructions in performance-critical operations:  AVX2 AVX512F FMA
To enable them in other operations, rebuild TensorFlow with the appropriate compiler flags.
 iteration 0: train_error=1.000227, test_error=1.000253
 iteration 10: train_error=0.998526, test_error=1.000226
 iteration 20: train_error=0.996735, test_error=1.000124
 iteration 30: train_error=0.994730, test_error=0.999840
 iteration 40: train_error=0.992304, test_error=0.999192
 iteration 50: train_error=0.989093, test_error=0.997844
 iteration 60: train_error=0.984439, test_error=0.995181
 iteration 70: train_error=0.977182, test_error=0.990101
 iteration 80: train_error=0.965441, test_error=0.980783
 iteration 90: train_error=0.946653, test_error=0.964688
 iteration 100: train_error=0.918552, test_error=0.939434
 iteration 110: train_error=0.881324, test_error=0.904881
 iteration 120: train_error=0.838524, test_error=0.864285
 iteration 130: train_error=0.794060, test_error=0.821610
 iteration 140: train_error=0.749451, test_error=0.778546
 iteration 150: train_error=0.705176, test_error=0.735538
 iteration 160: train_error=0.662150, test_error=0.693428
 iteration 170: train_error=0.621414, test_error=0.653329
 iteration 180: train_error=0.583612, test_error=0.616041
 iteration 190: train_error=0.548947, test_error=0.581904
 iteration 200: train_error=0.517337, test_error=0.550920
 iteration 210: train_error=0.488553, test_error=0.522895
 iteration 220: train_error=0.462309, test_error=0.497551
 iteration 230: train_error=0.438313, test_error=0.474591
 iteration 240: train_error=0.416298, test_error=0.453736
 iteration 250: train_error=0.396026, test_error=0.434737
 iteration 260: train_error=0.377290, test_error=0.417378
 iteration 270: train_error=0.359911, test_error=0.401471
 iteration 280: train_error=0.343738, test_error=0.386853
 iteration 290: train_error=0.328637, test_error=0.373384
 iteration 300: train_error=0.314496, test_error=0.360940
 iteration 310: train_error=0.301218, test_error=0.349415
 iteration 320: train_error=0.288720, test_error=0.338716
 iteration 330: train_error=0.276931, test_error=0.328762
 iteration 340: train_error=0.265788, test_error=0.319483
 iteration 350: train_error=0.255239, test_error=0.310817
 iteration 360: train_error=0.245237, test_error=0.302710
 iteration 370: train_error=0.235739, test_error=0.295116
 iteration 380: train_error=0.226710, test_error=0.287991
 iteration 390: train_error=0.218114, test_error=0.281299
 iteration 400: train_error=0.209923, test_error=0.275006
 iteration 410: train_error=0.202108, test_error=0.269081
 iteration 420: train_error=0.194645, test_error=0.263498
 iteration 430: train_error=0.187509, test_error=0.258230
 iteration 440: train_error=0.180679, test_error=0.253255
 iteration 450: train_error=0.174136, test_error=0.248552
 iteration 460: train_error=0.167861, test_error=0.244101
 iteration 470: train_error=0.161839, test_error=0.239885
 iteration 480: train_error=0.156052, test_error=0.235888
 iteration 490: train_error=0.150488, test_error=0.232094
 iteration 500: train_error=0.145133, test_error=0.228489
 iteration 510: train_error=0.139977, test_error=0.225062
 iteration 520: train_error=0.135007, test_error=0.221801
 iteration 530: train_error=0.130215, test_error=0.218694
 iteration 540: train_error=0.125591, test_error=0.215732
 iteration 550: train_error=0.121127, test_error=0.212907
 iteration 560: train_error=0.116817, test_error=0.210210
 iteration 570: train_error=0.112652, test_error=0.207633
 iteration 580: train_error=0.108628, test_error=0.205169
 iteration 590: train_error=0.104738, test_error=0.202813
 iteration 600: train_error=0.100977, test_error=0.200557
 iteration 610: train_error=0.097341, test_error=0.198396
 iteration 620: train_error=0.093825, test_error=0.196326
 iteration 630: train_error=0.090425, test_error=0.194341
 iteration 640: train_error=0.087137, test_error=0.192437
 iteration 650: train_error=0.083958, test_error=0.190610
 iteration 660: train_error=0.080884, test_error=0.188856
 iteration 670: train_error=0.077913, test_error=0.187172
 iteration 680: train_error=0.075042, test_error=0.185554
 iteration 690: train_error=0.072267, test_error=0.183999
 iteration 700: train_error=0.069587, test_error=0.182503
 iteration 710: train_error=0.066998, test_error=0.181066
 iteration 720: train_error=0.064499, test_error=0.179683
 iteration 730: train_error=0.062086, test_error=0.178352
 iteration 740: train_error=0.059758, test_error=0.177071
 iteration 750: train_error=0.057513, test_error=0.175839
 iteration 760: train_error=0.055348, test_error=0.174652
 iteration 770: train_error=0.053261, test_error=0.173508
 iteration 780: train_error=0.051250, test_error=0.172407
 iteration 790: train_error=0.049313, test_error=0.171346
 iteration 800: train_error=0.047447, test_error=0.170324
 iteration 810: train_error=0.045651, test_error=0.169339
 iteration 820: train_error=0.043923, test_error=0.168389
 iteration 830: train_error=0.042260, test_error=0.167473
 iteration 840: train_error=0.040661, test_error=0.166590
 iteration 850: train_error=0.039123, test_error=0.165739
 iteration 860: train_error=0.037645, test_error=0.164918
 iteration 870: train_error=0.036225, test_error=0.164126
 iteration 880: train_error=0.034860, test_error=0.163361
 iteration 890: train_error=0.033549, test_error=0.162624
 iteration 900: train_error=0.032289, test_error=0.161912
 iteration 910: train_error=0.031080, test_error=0.161225
 iteration 920: train_error=0.029919, test_error=0.160562
 iteration 930: train_error=0.028804, test_error=0.159922
 iteration 940: train_error=0.027733, test_error=0.159304
 iteration 950: train_error=0.026706, test_error=0.158707
 iteration 960: train_error=0.025719, test_error=0.158130
 iteration 970: train_error=0.024772, test_error=0.157574
 iteration 980: train_error=0.023863, test_error=0.157036
 iteration 990: train_error=0.022991, test_error=0.156516
 iteration 1000: train_error=0.022153, test_error=0.156014
 iteration 1010: train_error=0.021349, test_error=0.155528
 iteration 1020: train_error=0.020578, test_error=0.155059
 iteration 1030: train_error=0.019837, test_error=0.154605
 iteration 1040: train_error=0.019125, test_error=0.154167
 iteration 1050: train_error=0.018442, test_error=0.153743
 iteration 1060: train_error=0.017786, test_error=0.153332
 iteration 1070: train_error=0.017156, test_error=0.152936
 iteration 1080: train_error=0.016551, test_error=0.152552
 iteration 1090: train_error=0.015969, test_error=0.152181
 iteration 1100: train_error=0.015411, test_error=0.151821
 iteration 1110: train_error=0.014874, test_error=0.151474
 iteration 1120: train_error=0.014358, test_error=0.151137
 iteration 1130: train_error=0.013862, test_error=0.150811
 iteration 1140: train_error=0.013386, test_error=0.150495
 iteration 1150: train_error=0.012928, test_error=0.150190
 iteration 1160: train_error=0.012487, test_error=0.149894
 iteration 1170: train_error=0.012064, test_error=0.149607
 iteration 1180: train_error=0.011656, test_error=0.149330
 iteration 1190: train_error=0.011265, test_error=0.149061
 iteration 1200: train_error=0.010888, test_error=0.148800
 iteration 1210: train_error=0.010525, test_error=0.148547
 iteration 1220: train_error=0.010176, test_error=0.148302
 iteration 1230: train_error=0.009840, test_error=0.148065
 iteration 1240: train_error=0.009517, test_error=0.147834
 iteration 1250: train_error=0.009206, test_error=0.147611
 iteration 1260: train_error=0.008906, test_error=0.147395
 iteration 1270: train_error=0.008618, test_error=0.147184
 iteration 1280: train_error=0.008340, test_error=0.146981
 iteration 1290: train_error=0.008073, test_error=0.146783
 iteration 1300: train_error=0.007815, test_error=0.146591
 iteration 1310: train_error=0.007567, test_error=0.146404
 iteration 1320: train_error=0.007328, test_error=0.146224
 iteration 1330: train_error=0.007097, test_error=0.146048
 iteration 1340: train_error=0.006875, test_error=0.145877
 iteration 1350: train_error=0.006661, test_error=0.145712
 iteration 1360: train_error=0.006455, test_error=0.145551
 iteration 1370: train_error=0.006256, test_error=0.145395
 iteration 1380: train_error=0.006064, test_error=0.145243
 iteration 1390: train_error=0.005879, test_error=0.145095
 iteration 1400: train_error=0.005701, test_error=0.144952
 iteration 1410: train_error=0.005529, test_error=0.144813
 iteration 1420: train_error=0.005363, test_error=0.144677
 iteration 1430: train_error=0.005203, test_error=0.144545
 iteration 1440: train_error=0.005049, test_error=0.144417
 iteration 1450: train_error=0.004900, test_error=0.144293
 iteration 1460: train_error=0.004756, test_error=0.144171
 iteration 1470: train_error=0.004618, test_error=0.144054
 iteration 1480: train_error=0.004484, test_error=0.143939
 iteration 1490: train_error=0.004355, test_error=0.143827
 iteration 1500: train_error=0.004230, test_error=0.143718
 iteration 1510: train_error=0.004110, test_error=0.143613
 iteration 1520: train_error=0.003993, test_error=0.143510
 iteration 1530: train_error=0.003881, test_error=0.143409
 iteration 1540: train_error=0.003773, test_error=0.143312
 iteration 1550: train_error=0.003668, test_error=0.143217
 iteration 1560: train_error=0.003567, test_error=0.143124
 iteration 1570: train_error=0.003469, test_error=0.143034
 iteration 1580: train_error=0.003374, test_error=0.142946
 iteration 1590: train_error=0.003283, test_error=0.142860
 iteration 1600: train_error=0.003195, test_error=0.142776
 iteration 1610: train_error=0.003110, test_error=0.142695
 iteration 1620: train_error=0.003027, test_error=0.142616
 iteration 1630: train_error=0.002948, test_error=0.142538
 iteration 1640: train_error=0.002871, test_error=0.142463
 iteration 1650: train_error=0.002796, test_error=0.142389
 iteration 1660: train_error=0.002724, test_error=0.142317
 iteration 1670: train_error=0.002654, test_error=0.142247
 iteration 1680: train_error=0.002587, test_error=0.142179
 iteration 1690: train_error=0.002522, test_error=0.142112
 iteration 1700: train_error=0.002459, test_error=0.142047
 iteration 1710: train_error=0.002398, test_error=0.141983
 iteration 1720: train_error=0.002338, test_error=0.141921
 iteration 1730: train_error=0.002281, test_error=0.141860
 iteration 1740: train_error=0.002226, test_error=0.141801
 iteration 1750: train_error=0.002172, test_error=0.141743
 iteration 1760: train_error=0.002120, test_error=0.141687
 iteration 1770: train_error=0.002070, test_error=0.141632
 iteration 1780: train_error=0.002022, test_error=0.141578
 iteration 1790: train_error=0.001974, test_error=0.141525
 iteration 1800: train_error=0.001929, test_error=0.141473
 iteration 1810: train_error=0.001885, test_error=0.141423
 iteration 1820: train_error=0.001842, test_error=0.141374
 iteration 1830: train_error=0.001800, test_error=0.141326
 iteration 1840: train_error=0.001760, test_error=0.141278
 iteration 1850: train_error=0.001721, test_error=0.141232
 iteration 1860: train_error=0.001683, test_error=0.141187
 iteration 1870: train_error=0.001646, test_error=0.141143
 iteration 1880: train_error=0.001611, test_error=0.141100
 iteration 1890: train_error=0.001576, test_error=0.141058
 iteration 1900: train_error=0.001543, test_error=0.141017
 iteration 1910: train_error=0.001510, test_error=0.140976
 iteration 1920: train_error=0.001479, test_error=0.140937
 iteration 1930: train_error=0.001448, test_error=0.140898
 iteration 1940: train_error=0.001419, test_error=0.140860
 iteration 1950: train_error=0.001390, test_error=0.140823
 iteration 1960: train_error=0.001362, test_error=0.140786
 iteration 1970: train_error=0.001335, test_error=0.140750
 iteration 1980: train_error=0.001309, test_error=0.140716
 iteration 1990: train_error=0.001283, test_error=0.140681
 iteration 2000: train_error=0.001258, test_error=0.140648
[{'train_error': 0.001258304, 'test_error': 0.14064766}]
_images/Collaborative_filtering_27_203.png

Regularized moodel

reg_model = build_matrix_norm.build_reg_model(regularization_coeff=0.1, gravity_coeff=1.0, embedding_dim=35,init_stddev=.05)
reg_model.train(num_iterations=2000, learning_rate=20.)
 iteration 0: train_error_observed=1.000023, test_error_observed=0.999826, observed_loss=1.000023, regularization_loss=0.017468, gravity_loss=0.000218
 iteration 10: train_error_observed=0.998364, test_error_observed=0.999792, observed_loss=0.998364, regularization_loss=0.017070, gravity_loss=0.000208
 iteration 20: train_error_observed=0.996703, test_error_observed=0.999693, observed_loss=0.996703, regularization_loss=0.016727, gravity_loss=0.000200
 iteration 30: train_error_observed=0.994921, test_error_observed=0.999432, observed_loss=0.994921, regularization_loss=0.016436, gravity_loss=0.000192
 iteration 40: train_error_observed=0.992834, test_error_observed=0.998848, observed_loss=0.992834, regularization_loss=0.016200, gravity_loss=0.000187
 iteration 50: train_error_observed=0.990130, test_error_observed=0.997656, observed_loss=0.990130, regularization_loss=0.016027, gravity_loss=0.000183
 iteration 60: train_error_observed=0.986250, test_error_observed=0.995338, observed_loss=0.986250, regularization_loss=0.015933, gravity_loss=0.000180
 iteration 70: train_error_observed=0.980224, test_error_observed=0.990973, observed_loss=0.980224, regularization_loss=0.015951, gravity_loss=0.000181
 iteration 80: train_error_observed=0.970457, test_error_observed=0.983033, observed_loss=0.970457, regularization_loss=0.016139, gravity_loss=0.000188
 iteration 90: train_error_observed=0.954685, test_error_observed=0.969305, observed_loss=0.954685, regularization_loss=0.016587, gravity_loss=0.000206
 iteration 100: train_error_observed=0.930580, test_error_observed=0.947421, observed_loss=0.930580, regularization_loss=0.017418, gravity_loss=0.000247
 iteration 110: train_error_observed=0.897485, test_error_observed=0.916504, observed_loss=0.897485, regularization_loss=0.018743, gravity_loss=0.000334
 iteration 120: train_error_observed=0.857936, test_error_observed=0.878784, observed_loss=0.857936, regularization_loss=0.020579, gravity_loss=0.000501
 iteration 130: train_error_observed=0.816035, test_error_observed=0.838298, observed_loss=0.816035, regularization_loss=0.022820, gravity_loss=0.000774
 iteration 140: train_error_observed=0.774204, test_error_observed=0.797634, observed_loss=0.774204, regularization_loss=0.025316, gravity_loss=0.001170
 iteration 150: train_error_observed=0.733110, test_error_observed=0.757526, observed_loss=0.733110, regularization_loss=0.027972, gravity_loss=0.001697
 iteration 160: train_error_observed=0.693308, test_error_observed=0.718463, observed_loss=0.693308, regularization_loss=0.030736, gravity_loss=0.002367
 iteration 170: train_error_observed=0.655570, test_error_observed=0.681216, observed_loss=0.655570, regularization_loss=0.033558, gravity_loss=0.003184
 iteration 180: train_error_observed=0.620484, test_error_observed=0.646454, observed_loss=0.620484, regularization_loss=0.036381, gravity_loss=0.004142
 iteration 190: train_error_observed=0.588288, test_error_observed=0.614522, observed_loss=0.588288, regularization_loss=0.039155, gravity_loss=0.005227
 iteration 200: train_error_observed=0.558957, test_error_observed=0.585471, observed_loss=0.558957, regularization_loss=0.041842, gravity_loss=0.006420
 iteration 210: train_error_observed=0.532322, test_error_observed=0.559171, observed_loss=0.532322, regularization_loss=0.044417, gravity_loss=0.007703
 iteration 220: train_error_observed=0.508147, test_error_observed=0.535407, observed_loss=0.508147, regularization_loss=0.046867, gravity_loss=0.009057
 iteration 230: train_error_observed=0.486184, test_error_observed=0.513930, observed_loss=0.486184, regularization_loss=0.049187, gravity_loss=0.010466
 iteration 240: train_error_observed=0.466191, test_error_observed=0.494496, observed_loss=0.466191, regularization_loss=0.051377, gravity_loss=0.011915
 iteration 250: train_error_observed=0.447949, test_error_observed=0.476880, observed_loss=0.447949, regularization_loss=0.053439, gravity_loss=0.013392
 iteration 260: train_error_observed=0.431263, test_error_observed=0.460881, observed_loss=0.431263, regularization_loss=0.055378, gravity_loss=0.014885
 iteration 270: train_error_observed=0.415961, test_error_observed=0.446321, observed_loss=0.415961, regularization_loss=0.057200, gravity_loss=0.016386
 iteration 280: train_error_observed=0.401893, test_error_observed=0.433043, observed_loss=0.401893, regularization_loss=0.058910, gravity_loss=0.017886
 iteration 290: train_error_observed=0.388928, test_error_observed=0.420911, observed_loss=0.388928, regularization_loss=0.060515, gravity_loss=0.019379
 iteration 300: train_error_observed=0.376949, test_error_observed=0.409802, observed_loss=0.376949, regularization_loss=0.062021, gravity_loss=0.020858
 iteration 310: train_error_observed=0.365857, test_error_observed=0.399610, observed_loss=0.365857, regularization_loss=0.063433, gravity_loss=0.022318
 iteration 320: train_error_observed=0.355564, test_error_observed=0.390240, observed_loss=0.355564, regularization_loss=0.064759, gravity_loss=0.023756
 iteration 330: train_error_observed=0.345990, test_error_observed=0.381608, observed_loss=0.345990, regularization_loss=0.066002, gravity_loss=0.025167
 iteration 340: train_error_observed=0.337068, test_error_observed=0.373640, observed_loss=0.337068, regularization_loss=0.067169, gravity_loss=0.026549
 iteration 350: train_error_observed=0.328738, test_error_observed=0.366272, observed_loss=0.328738, regularization_loss=0.068264, gravity_loss=0.027900
 iteration 360: train_error_observed=0.320945, test_error_observed=0.359446, observed_loss=0.320945, regularization_loss=0.069292, gravity_loss=0.029217
 iteration 370: train_error_observed=0.313642, test_error_observed=0.353111, observed_loss=0.313642, regularization_loss=0.070257, gravity_loss=0.030498
 iteration 380: train_error_observed=0.306788, test_error_observed=0.347222, observed_loss=0.306788, regularization_loss=0.071165, gravity_loss=0.031744
 iteration 390: train_error_observed=0.300343, test_error_observed=0.341739, observed_loss=0.300343, regularization_loss=0.072018, gravity_loss=0.032952
 iteration 400: train_error_observed=0.294275, test_error_observed=0.336627, observed_loss=0.294275, regularization_loss=0.072821, gravity_loss=0.034123
 iteration 410: train_error_observed=0.288553, test_error_observed=0.331853, observed_loss=0.288553, regularization_loss=0.073577, gravity_loss=0.035256
 iteration 420: train_error_observed=0.283150, test_error_observed=0.327390, observed_loss=0.283150, regularization_loss=0.074288, gravity_loss=0.036350
 iteration 430: train_error_observed=0.278040, test_error_observed=0.323212, observed_loss=0.278040, regularization_loss=0.074960, gravity_loss=0.037407
 iteration 440: train_error_observed=0.273203, test_error_observed=0.319297, observed_loss=0.273203, regularization_loss=0.075593, gravity_loss=0.038425
 iteration 450: train_error_observed=0.268616, test_error_observed=0.315622, observed_loss=0.268616, regularization_loss=0.076191, gravity_loss=0.039406
 iteration 460: train_error_observed=0.264261, test_error_observed=0.312171, observed_loss=0.264261, regularization_loss=0.076756, gravity_loss=0.040350
 iteration 470: train_error_observed=0.260123, test_error_observed=0.308926, observed_loss=0.260123, regularization_loss=0.077292, gravity_loss=0.041257
 iteration 480: train_error_observed=0.256184, test_error_observed=0.305871, observed_loss=0.256184, regularization_loss=0.077799, gravity_loss=0.042128
 iteration 490: train_error_observed=0.252430, test_error_observed=0.302993, observed_loss=0.252430, regularization_loss=0.078281, gravity_loss=0.042964
 iteration 500: train_error_observed=0.248849, test_error_observed=0.300278, observed_loss=0.248849, regularization_loss=0.078738, gravity_loss=0.043765
 iteration 510: train_error_observed=0.245429, test_error_observed=0.297715, observed_loss=0.245429, regularization_loss=0.079174, gravity_loss=0.044532
 iteration 520: train_error_observed=0.242157, test_error_observed=0.295293, observed_loss=0.242157, regularization_loss=0.079589, gravity_loss=0.045266
 iteration 530: train_error_observed=0.239025, test_error_observed=0.293002, observed_loss=0.239025, regularization_loss=0.079986, gravity_loss=0.045969
 iteration 540: train_error_observed=0.236021, test_error_observed=0.290832, observed_loss=0.236021, regularization_loss=0.080366, gravity_loss=0.046639
 iteration 550: train_error_observed=0.233138, test_error_observed=0.288777, observed_loss=0.233138, regularization_loss=0.080730, gravity_loss=0.047280
 iteration 560: train_error_observed=0.230368, test_error_observed=0.286827, observed_loss=0.230368, regularization_loss=0.081080, gravity_loss=0.047891
 iteration 570: train_error_observed=0.227703, test_error_observed=0.284976, observed_loss=0.227703, regularization_loss=0.081417, gravity_loss=0.048474
 iteration 580: train_error_observed=0.225135, test_error_observed=0.283217, observed_loss=0.225135, regularization_loss=0.081742, gravity_loss=0.049029
 iteration 590: train_error_observed=0.222659, test_error_observed=0.281544, observed_loss=0.222659, regularization_loss=0.082057, gravity_loss=0.049557
 iteration 600: train_error_observed=0.220269, test_error_observed=0.279952, observed_loss=0.220269, regularization_loss=0.082362, gravity_loss=0.050060
 iteration 610: train_error_observed=0.217959, test_error_observed=0.278434, observed_loss=0.217959, regularization_loss=0.082659, gravity_loss=0.050537
 iteration 620: train_error_observed=0.215723, test_error_observed=0.276988, observed_loss=0.215723, regularization_loss=0.082948, gravity_loss=0.050991
 iteration 630: train_error_observed=0.213558, test_error_observed=0.275607, observed_loss=0.213558, regularization_loss=0.083230, gravity_loss=0.051421
 iteration 640: train_error_observed=0.211459, test_error_observed=0.274289, observed_loss=0.211459, regularization_loss=0.083506, gravity_loss=0.051829
 iteration 650: train_error_observed=0.209421, test_error_observed=0.273028, observed_loss=0.209421, regularization_loss=0.083777, gravity_loss=0.052216
 iteration 660: train_error_observed=0.207441, test_error_observed=0.271822, observed_loss=0.207441, regularization_loss=0.084043, gravity_loss=0.052582
 iteration 670: train_error_observed=0.205515, test_error_observed=0.270668, observed_loss=0.205515, regularization_loss=0.084305, gravity_loss=0.052928
 iteration 680: train_error_observed=0.203640, test_error_observed=0.269562, observed_loss=0.203640, regularization_loss=0.084564, gravity_loss=0.053254
 iteration 690: train_error_observed=0.201813, test_error_observed=0.268501, observed_loss=0.201813, regularization_loss=0.084819, gravity_loss=0.053562
 iteration 700: train_error_observed=0.200031, test_error_observed=0.267483, observed_loss=0.200031, regularization_loss=0.085073, gravity_loss=0.053852
 iteration 710: train_error_observed=0.198292, test_error_observed=0.266505, observed_loss=0.198292, regularization_loss=0.085324, gravity_loss=0.054125
 iteration 720: train_error_observed=0.196593, test_error_observed=0.265566, observed_loss=0.196593, regularization_loss=0.085574, gravity_loss=0.054382
 iteration 730: train_error_observed=0.194931, test_error_observed=0.264662, observed_loss=0.194931, regularization_loss=0.085823, gravity_loss=0.054623
 iteration 740: train_error_observed=0.193305, test_error_observed=0.263793, observed_loss=0.193305, regularization_loss=0.086071, gravity_loss=0.054848
 iteration 750: train_error_observed=0.191713, test_error_observed=0.262955, observed_loss=0.191713, regularization_loss=0.086318, gravity_loss=0.055059
 iteration 760: train_error_observed=0.190152, test_error_observed=0.262148, observed_loss=0.190152, regularization_loss=0.086565, gravity_loss=0.055256
 iteration 770: train_error_observed=0.188622, test_error_observed=0.261370, observed_loss=0.188622, regularization_loss=0.086813, gravity_loss=0.055439
 iteration 780: train_error_observed=0.187120, test_error_observed=0.260619, observed_loss=0.187120, regularization_loss=0.087060, gravity_loss=0.055609
 iteration 790: train_error_observed=0.185645, test_error_observed=0.259895, observed_loss=0.185645, regularization_loss=0.087308, gravity_loss=0.055766
 iteration 800: train_error_observed=0.184196, test_error_observed=0.259194, observed_loss=0.184196, regularization_loss=0.087556, gravity_loss=0.055912
 iteration 810: train_error_observed=0.182772, test_error_observed=0.258517, observed_loss=0.182772, regularization_loss=0.087806, gravity_loss=0.056046
 iteration 820: train_error_observed=0.181370, test_error_observed=0.257863, observed_loss=0.181370, regularization_loss=0.088056, gravity_loss=0.056169
 iteration 830: train_error_observed=0.179991, test_error_observed=0.257229, observed_loss=0.179991, regularization_loss=0.088307, gravity_loss=0.056281
 iteration 840: train_error_observed=0.178633, test_error_observed=0.256616, observed_loss=0.178633, regularization_loss=0.088560, gravity_loss=0.056383
 iteration 850: train_error_observed=0.177295, test_error_observed=0.256021, observed_loss=0.177295, regularization_loss=0.088814, gravity_loss=0.056475
 iteration 860: train_error_observed=0.175977, test_error_observed=0.255446, observed_loss=0.175977, regularization_loss=0.089069, gravity_loss=0.056558
 iteration 870: train_error_observed=0.174677, test_error_observed=0.254887, observed_loss=0.174677, regularization_loss=0.089326, gravity_loss=0.056632
 iteration 880: train_error_observed=0.173395, test_error_observed=0.254345, observed_loss=0.173395, regularization_loss=0.089583, gravity_loss=0.056697
 iteration 890: train_error_observed=0.172130, test_error_observed=0.253820, observed_loss=0.172130, regularization_loss=0.089843, gravity_loss=0.056754
 iteration 900: train_error_observed=0.170882, test_error_observed=0.253309, observed_loss=0.170882, regularization_loss=0.090104, gravity_loss=0.056802
 iteration 910: train_error_observed=0.169650, test_error_observed=0.252813, observed_loss=0.169650, regularization_loss=0.090366, gravity_loss=0.056843
 iteration 920: train_error_observed=0.168433, test_error_observed=0.252331, observed_loss=0.168433, regularization_loss=0.090630, gravity_loss=0.056877
 iteration 930: train_error_observed=0.167231, test_error_observed=0.251863, observed_loss=0.167231, regularization_loss=0.090895, gravity_loss=0.056904
 iteration 940: train_error_observed=0.166044, test_error_observed=0.251408, observed_loss=0.166044, regularization_loss=0.091162, gravity_loss=0.056924
 iteration 950: train_error_observed=0.164870, test_error_observed=0.250964, observed_loss=0.164870, regularization_loss=0.091430, gravity_loss=0.056937
 iteration 960: train_error_observed=0.163711, test_error_observed=0.250533, observed_loss=0.163711, regularization_loss=0.091700, gravity_loss=0.056944
 iteration 970: train_error_observed=0.162564, test_error_observed=0.250113, observed_loss=0.162564, regularization_loss=0.091971, gravity_loss=0.056946
 iteration 980: train_error_observed=0.161431, test_error_observed=0.249705, observed_loss=0.161431, regularization_loss=0.092243, gravity_loss=0.056941
 iteration 990: train_error_observed=0.160310, test_error_observed=0.249306, observed_loss=0.160310, regularization_loss=0.092516, gravity_loss=0.056932
 iteration 1000: train_error_observed=0.159202, test_error_observed=0.248918, observed_loss=0.159202, regularization_loss=0.092791, gravity_loss=0.056917
 iteration 1010: train_error_observed=0.158106, test_error_observed=0.248540, observed_loss=0.158106, regularization_loss=0.093066, gravity_loss=0.056897
 iteration 1020: train_error_observed=0.157022, test_error_observed=0.248172, observed_loss=0.157022, regularization_loss=0.093343, gravity_loss=0.056872
 iteration 1030: train_error_observed=0.155950, test_error_observed=0.247812, observed_loss=0.155950, regularization_loss=0.093621, gravity_loss=0.056843
 iteration 1040: train_error_observed=0.154889, test_error_observed=0.247462, observed_loss=0.154889, regularization_loss=0.093899, gravity_loss=0.056810
 iteration 1050: train_error_observed=0.153840, test_error_observed=0.247120, observed_loss=0.153840, regularization_loss=0.094179, gravity_loss=0.056772
 iteration 1060: train_error_observed=0.152802, test_error_observed=0.246786, observed_loss=0.152802, regularization_loss=0.094459, gravity_loss=0.056731
 iteration 1070: train_error_observed=0.151775, test_error_observed=0.246460, observed_loss=0.151775, regularization_loss=0.094739, gravity_loss=0.056686
 iteration 1080: train_error_observed=0.150759, test_error_observed=0.246142, observed_loss=0.150759, regularization_loss=0.095020, gravity_loss=0.056637
 iteration 1090: train_error_observed=0.149754, test_error_observed=0.245832, observed_loss=0.149754, regularization_loss=0.095302, gravity_loss=0.056585
 iteration 1100: train_error_observed=0.148760, test_error_observed=0.245529, observed_loss=0.148760, regularization_loss=0.095584, gravity_loss=0.056530
 iteration 1110: train_error_observed=0.147776, test_error_observed=0.245233, observed_loss=0.147776, regularization_loss=0.095866, gravity_loss=0.056471
 iteration 1120: train_error_observed=0.146803, test_error_observed=0.244944, observed_loss=0.146803, regularization_loss=0.096148, gravity_loss=0.056410
 iteration 1130: train_error_observed=0.145840, test_error_observed=0.244661, observed_loss=0.145840, regularization_loss=0.096431, gravity_loss=0.056346
 iteration 1140: train_error_observed=0.144887, test_error_observed=0.244385, observed_loss=0.144887, regularization_loss=0.096713, gravity_loss=0.056279
 iteration 1150: train_error_observed=0.143945, test_error_observed=0.244115, observed_loss=0.143945, regularization_loss=0.096995, gravity_loss=0.056210
 iteration 1160: train_error_observed=0.143013, test_error_observed=0.243851, observed_loss=0.143013, regularization_loss=0.097278, gravity_loss=0.056139
 iteration 1170: train_error_observed=0.142091, test_error_observed=0.243594, observed_loss=0.142091, regularization_loss=0.097559, gravity_loss=0.056065
 iteration 1180: train_error_observed=0.141179, test_error_observed=0.243342, observed_loss=0.141179, regularization_loss=0.097841, gravity_loss=0.055989
 iteration 1190: train_error_observed=0.140277, test_error_observed=0.243095, observed_loss=0.140277, regularization_loss=0.098122, gravity_loss=0.055911
 iteration 1200: train_error_observed=0.139385, test_error_observed=0.242854, observed_loss=0.139385, regularization_loss=0.098403, gravity_loss=0.055832
 iteration 1210: train_error_observed=0.138502, test_error_observed=0.242619, observed_loss=0.138502, regularization_loss=0.098682, gravity_loss=0.055751
 iteration 1220: train_error_observed=0.137629, test_error_observed=0.242388, observed_loss=0.137629, regularization_loss=0.098962, gravity_loss=0.055668
 iteration 1230: train_error_observed=0.136766, test_error_observed=0.242163, observed_loss=0.136766, regularization_loss=0.099240, gravity_loss=0.055583
 iteration 1240: train_error_observed=0.135912, test_error_observed=0.241942, observed_loss=0.135912, regularization_loss=0.099518, gravity_loss=0.055497
 iteration 1250: train_error_observed=0.135068, test_error_observed=0.241727, observed_loss=0.135068, regularization_loss=0.099795, gravity_loss=0.055410
 iteration 1260: train_error_observed=0.134233, test_error_observed=0.241515, observed_loss=0.134233, regularization_loss=0.100070, gravity_loss=0.055321
 iteration 1270: train_error_observed=0.133408, test_error_observed=0.241309, observed_loss=0.133408, regularization_loss=0.100345, gravity_loss=0.055232
 iteration 1280: train_error_observed=0.132591, test_error_observed=0.241107, observed_loss=0.132591, regularization_loss=0.100619, gravity_loss=0.055141
 iteration 1290: train_error_observed=0.131784, test_error_observed=0.240909, observed_loss=0.131784, regularization_loss=0.100891, gravity_loss=0.055049
 iteration 1300: train_error_observed=0.130986, test_error_observed=0.240716, observed_loss=0.130986, regularization_loss=0.101163, gravity_loss=0.054956
 iteration 1310: train_error_observed=0.130197, test_error_observed=0.240526, observed_loss=0.130197, regularization_loss=0.101433, gravity_loss=0.054863
 iteration 1320: train_error_observed=0.129417, test_error_observed=0.240341, observed_loss=0.129417, regularization_loss=0.101701, gravity_loss=0.054768
 iteration 1330: train_error_observed=0.128646, test_error_observed=0.240159, observed_loss=0.128646, regularization_loss=0.101969, gravity_loss=0.054673
 iteration 1340: train_error_observed=0.127883, test_error_observed=0.239982, observed_loss=0.127883, regularization_loss=0.102234, gravity_loss=0.054577
 iteration 1350: train_error_observed=0.127129, test_error_observed=0.239808, observed_loss=0.127129, regularization_loss=0.102499, gravity_loss=0.054481
 iteration 1360: train_error_observed=0.126384, test_error_observed=0.239638, observed_loss=0.126384, regularization_loss=0.102762, gravity_loss=0.054384
 iteration 1370: train_error_observed=0.125647, test_error_observed=0.239471, observed_loss=0.125647, regularization_loss=0.103023, gravity_loss=0.054287
 iteration 1380: train_error_observed=0.124919, test_error_observed=0.239308, observed_loss=0.124919, regularization_loss=0.103283, gravity_loss=0.054189
 iteration 1390: train_error_observed=0.124199, test_error_observed=0.239148, observed_loss=0.124199, regularization_loss=0.103541, gravity_loss=0.054091
 iteration 1400: train_error_observed=0.123488, test_error_observed=0.238992, observed_loss=0.123488, regularization_loss=0.103798, gravity_loss=0.053992
 iteration 1410: train_error_observed=0.122784, test_error_observed=0.238838, observed_loss=0.122784, regularization_loss=0.104052, gravity_loss=0.053894
 iteration 1420: train_error_observed=0.122089, test_error_observed=0.238688, observed_loss=0.122089, regularization_loss=0.104306, gravity_loss=0.053795
 iteration 1430: train_error_observed=0.121401, test_error_observed=0.238542, observed_loss=0.121401, regularization_loss=0.104557, gravity_loss=0.053695
 iteration 1440: train_error_observed=0.120722, test_error_observed=0.238398, observed_loss=0.120722, regularization_loss=0.104807, gravity_loss=0.053596
 iteration 1450: train_error_observed=0.120050, test_error_observed=0.238257, observed_loss=0.120050, regularization_loss=0.105055, gravity_loss=0.053497
 iteration 1460: train_error_observed=0.119386, test_error_observed=0.238119, observed_loss=0.119386, regularization_loss=0.105301, gravity_loss=0.053397
 iteration 1470: train_error_observed=0.118730, test_error_observed=0.237984, observed_loss=0.118730, regularization_loss=0.105545, gravity_loss=0.053298
 iteration 1480: train_error_observed=0.118081, test_error_observed=0.237851, observed_loss=0.118081, regularization_loss=0.105787, gravity_loss=0.053198
 iteration 1490: train_error_observed=0.117440, test_error_observed=0.237722, observed_loss=0.117440, regularization_loss=0.106028, gravity_loss=0.053099
 iteration 1500: train_error_observed=0.116806, test_error_observed=0.237595, observed_loss=0.116806, regularization_loss=0.106267, gravity_loss=0.052999
 iteration 1510: train_error_observed=0.116179, test_error_observed=0.237470, observed_loss=0.116179, regularization_loss=0.106504, gravity_loss=0.052900
 iteration 1520: train_error_observed=0.115560, test_error_observed=0.237348, observed_loss=0.115560, regularization_loss=0.106739, gravity_loss=0.052801
 iteration 1530: train_error_observed=0.114948, test_error_observed=0.237229, observed_loss=0.114948, regularization_loss=0.106972, gravity_loss=0.052702
 iteration 1540: train_error_observed=0.114343, test_error_observed=0.237112, observed_loss=0.114343, regularization_loss=0.107204, gravity_loss=0.052603
 iteration 1550: train_error_observed=0.113744, test_error_observed=0.236997, observed_loss=0.113744, regularization_loss=0.107433, gravity_loss=0.052504
 iteration 1560: train_error_observed=0.113153, test_error_observed=0.236885, observed_loss=0.113153, regularization_loss=0.107661, gravity_loss=0.052406
 iteration 1570: train_error_observed=0.112568, test_error_observed=0.236775, observed_loss=0.112568, regularization_loss=0.107887, gravity_loss=0.052308
 iteration 1580: train_error_observed=0.111990, test_error_observed=0.236667, observed_loss=0.111990, regularization_loss=0.108111, gravity_loss=0.052210
 iteration 1590: train_error_observed=0.111419, test_error_observed=0.236562, observed_loss=0.111419, regularization_loss=0.108333, gravity_loss=0.052112
 iteration 1600: train_error_observed=0.110854, test_error_observed=0.236458, observed_loss=0.110854, regularization_loss=0.108553, gravity_loss=0.052015
 iteration 1610: train_error_observed=0.110296, test_error_observed=0.236357, observed_loss=0.110296, regularization_loss=0.108771, gravity_loss=0.051918
 iteration 1620: train_error_observed=0.109744, test_error_observed=0.236258, observed_loss=0.109744, regularization_loss=0.108988, gravity_loss=0.051821
 iteration 1630: train_error_observed=0.109198, test_error_observed=0.236160, observed_loss=0.109198, regularization_loss=0.109202, gravity_loss=0.051725
 iteration 1640: train_error_observed=0.108659, test_error_observed=0.236065, observed_loss=0.108659, regularization_loss=0.109415, gravity_loss=0.051629
 iteration 1650: train_error_observed=0.108125, test_error_observed=0.235972, observed_loss=0.108125, regularization_loss=0.109626, gravity_loss=0.051534
 iteration 1660: train_error_observed=0.107598, test_error_observed=0.235880, observed_loss=0.107598, regularization_loss=0.109835, gravity_loss=0.051439
 iteration 1670: train_error_observed=0.107077, test_error_observed=0.235790, observed_loss=0.107077, regularization_loss=0.110043, gravity_loss=0.051344
 iteration 1680: train_error_observed=0.106561, test_error_observed=0.235703, observed_loss=0.106561, regularization_loss=0.110248, gravity_loss=0.051249
 iteration 1690: train_error_observed=0.106051, test_error_observed=0.235617, observed_loss=0.106051, regularization_loss=0.110452, gravity_loss=0.051156
 iteration 1700: train_error_observed=0.105547, test_error_observed=0.235532, observed_loss=0.105547, regularization_loss=0.110654, gravity_loss=0.051062
 iteration 1710: train_error_observed=0.105049, test_error_observed=0.235449, observed_loss=0.105049, regularization_loss=0.110854, gravity_loss=0.050969
 iteration 1720: train_error_observed=0.104556, test_error_observed=0.235369, observed_loss=0.104556, regularization_loss=0.111053, gravity_loss=0.050876
 iteration 1730: train_error_observed=0.104069, test_error_observed=0.235289, observed_loss=0.104069, regularization_loss=0.111249, gravity_loss=0.050784
 iteration 1740: train_error_observed=0.103587, test_error_observed=0.235211, observed_loss=0.103587, regularization_loss=0.111444, gravity_loss=0.050693
 iteration 1750: train_error_observed=0.103110, test_error_observed=0.235135, observed_loss=0.103110, regularization_loss=0.111638, gravity_loss=0.050601
 iteration 1760: train_error_observed=0.102639, test_error_observed=0.235061, observed_loss=0.102639, regularization_loss=0.111829, gravity_loss=0.050511
 iteration 1770: train_error_observed=0.102173, test_error_observed=0.234987, observed_loss=0.102173, regularization_loss=0.112019, gravity_loss=0.050421
 iteration 1780: train_error_observed=0.101712, test_error_observed=0.234916, observed_loss=0.101712, regularization_loss=0.112207, gravity_loss=0.050331
 iteration 1790: train_error_observed=0.101257, test_error_observed=0.234846, observed_loss=0.101257, regularization_loss=0.112394, gravity_loss=0.050242
 iteration 1800: train_error_observed=0.100806, test_error_observed=0.234777, observed_loss=0.100806, regularization_loss=0.112579, gravity_loss=0.050153
 iteration 1810: train_error_observed=0.100360, test_error_observed=0.234709, observed_loss=0.100360, regularization_loss=0.112762, gravity_loss=0.050065
 iteration 1820: train_error_observed=0.099919, test_error_observed=0.234643, observed_loss=0.099919, regularization_loss=0.112944, gravity_loss=0.049977
 iteration 1830: train_error_observed=0.099483, test_error_observed=0.234579, observed_loss=0.099483, regularization_loss=0.113124, gravity_loss=0.049890
 iteration 1840: train_error_observed=0.099051, test_error_observed=0.234515, observed_loss=0.099051, regularization_loss=0.113302, gravity_loss=0.049803
 iteration 1850: train_error_observed=0.098625, test_error_observed=0.234453, observed_loss=0.098625, regularization_loss=0.113479, gravity_loss=0.049717
 iteration 1860: train_error_observed=0.098202, test_error_observed=0.234392, observed_loss=0.098202, regularization_loss=0.113655, gravity_loss=0.049631
 iteration 1870: train_error_observed=0.097785, test_error_observed=0.234333, observed_loss=0.097785, regularization_loss=0.113829, gravity_loss=0.049546
 iteration 1880: train_error_observed=0.097372, test_error_observed=0.234274, observed_loss=0.097372, regularization_loss=0.114001, gravity_loss=0.049461
 iteration 1890: train_error_observed=0.096963, test_error_observed=0.234217, observed_loss=0.096963, regularization_loss=0.114172, gravity_loss=0.049377
 iteration 1900: train_error_observed=0.096559, test_error_observed=0.234161, observed_loss=0.096559, regularization_loss=0.114341, gravity_loss=0.049293
 iteration 1910: train_error_observed=0.096159, test_error_observed=0.234107, observed_loss=0.096159, regularization_loss=0.114509, gravity_loss=0.049210
 iteration 1920: train_error_observed=0.095763, test_error_observed=0.234053, observed_loss=0.095763, regularization_loss=0.114675, gravity_loss=0.049128
 iteration 1930: train_error_observed=0.095372, test_error_observed=0.234000, observed_loss=0.095372, regularization_loss=0.114840, gravity_loss=0.049046
 iteration 1940: train_error_observed=0.094985, test_error_observed=0.233949, observed_loss=0.094985, regularization_loss=0.115003, gravity_loss=0.048964
 iteration 1950: train_error_observed=0.094601, test_error_observed=0.233898, observed_loss=0.094601, regularization_loss=0.115165, gravity_loss=0.048883
 iteration 1960: train_error_observed=0.094222, test_error_observed=0.233849, observed_loss=0.094222, regularization_loss=0.115326, gravity_loss=0.048803
 iteration 1970: train_error_observed=0.093847, test_error_observed=0.233801, observed_loss=0.093847, regularization_loss=0.115485, gravity_loss=0.048723
 iteration 1980: train_error_observed=0.093476, test_error_observed=0.233754, observed_loss=0.093476, regularization_loss=0.115643, gravity_loss=0.048643
 iteration 1990: train_error_observed=0.093108, test_error_observed=0.233707, observed_loss=0.093108, regularization_loss=0.115800, gravity_loss=0.048564
 iteration 2000: train_error_observed=0.092745, test_error_observed=0.233662, observed_loss=0.092745, regularization_loss=0.115955, gravity_loss=0.048486
[{'train_error_observed': 0.09274472, 'test_error_observed': 0.23366201},
 {'observed_loss': 0.09274472,
  'regularization_loss': 0.11595495,
  'gravity_loss': 0.04848592}]
_images/Collaborative_filtering_30_202.png

In both models, we observe a steep loss in train error and test as the model progress. Although, the regularized model has a higher MSE, both on the training and test set. It must be noted that the quality of recommendation is improved when regularization is added, which is proven when the artist_neighbors() function is utilized. In addition, we observe in the end evaluation section, that the the performance of the model is improved when regularization is added. The test error decreases similarity to the test error, although it plateaus around the 1000 epoch mark. As expected, the the additional loss generated by the regularization functions increases over epochs. We add the following regularisation terms to our model.

  • Regularization of the model parameters. This is a common \(\ell_2\) regularization term on the embedding matrices, given by \(r(U, V) = \frac{1}{N} \sum_i \|U_i\|^2 + \frac{1}{M}\sum_j \|V_j\|^2\).

  • A global prior that pushes the prediction of any pair towards zero, called the gravity term. This is given by \(g(U, V) = \frac{1}{MN} \sum_{i = 1}^N \sum_{j = 1}^M \langle U_i, V_j \rangle^2\)

These terms modifies the “global” loss (as in, the sum of the network loss and the regularization loss) in order to drive the optimization algorithm in desired directions i.e. prevent overfitting.

Evaluating the embeddings

We will use two similairty meausres to inspect the robustness of our system:

  • Dot product: score of artist j \(\langle u, V_j \rangle\).

  • Cosine angle: score of artist j \(\frac{\langle u, V_j \rangle}{\|u\|\|V_j\|}\).

DOT = 'dot'
COSINE = 'cosine'
def compute_scores(query_embedding, item_embeddings, measure=DOT):
  """Computes the scores of the candidates given a query.
  Args:
    query_embedding: a vector of shape [k], representing the query embedding.
    item_embeddings: a matrix of shape [N, k], such that row i is the embedding
      of item i.
    measure: a string specifying the similarity measure to be used. Can be
      either DOT or COSINE.
  Returns:
    scores: a vector of shape [N], such that scores[i] is the score of item i.
  """
  u = query_embedding
  V = item_embeddings
  if measure == COSINE:
    V = V / np.linalg.norm(V, axis=1, keepdims=True)
    u = u / np.linalg.norm(u)
  scores = u.dot(V.T)
  return scores
def user_recommendations(model,user_id, k=15, measure=DOT, exclude_rated=False):
    scores = compute_scores(
        model.embeddings["userID"][user_id], model.embeddings["artistID"], measure)
    score_key = measure + ' score'
    df = pd.DataFrame({
        'score': list(scores),
        'name': artists.sort_values('artistID', ascending=True)['name'],
        'most assigned tag':artists.sort_values('artistID', ascending=True)['mostCommonGenre']
    })
    return df.sort_values(['score'], ascending=False).head(k)


def artist_neighbors(model, title_substring, measure=DOT, k=6):
  # Search for artist ids that match the given substring.
  inv_artist_id_mapping = {v: k for k, v in orginal_artist_ids.items()}
  ids =  artists[artists['name'].str.contains(title_substring)].artistID.values
  titles = artists[artists.artistID.isin(ids)]['name'].values
  if len(titles) == 0:
    raise ValueError("Found no artists with name %s" % title_substring)
  print("Nearest neighbors of : %s." % titles[0])
  if len(titles) > 1:
    print("[Found more than one matching artist. Other candidates: {}]".format(
        ", ".join(titles[1:])))
  artists_id_orginal = ids[0]
  asrtists_id_mapped = inv_artist_id_mapping[ids[0]]
  scores = compute_scores(
      model.embeddings["artistID"][asrtists_id_mapped], model.embeddings["artistID"],
      measure)
  score_key = measure + ' score'
  df = pd.DataFrame({
      score_key: list(scores),
      'name': artists.sort_values('artistID', ascending=True)['name'],
      'most assigned tag':artists.sort_values('artistID', ascending=True)['mostCommonGenre']
  })
  return df.sort_values([score_key], ascending=False).head(k)

Here, we find the most similar artists to the band the cure. We also include the most assigned tag associated with an artist. The reccomdations are conistent with our domain knowedge of bands similar to the cure.

artist_neighbors(vanilla_model, "The Cure", DOT)
Nearest neighbors of : The Cure.
dot score name most assigned tag
9437 0.529 The Cure chillout
17278 0.527 Kings of Leon chillout
58990 0.522 Queen 80's
3259 0.522 Coldplay chillout
26579 0.522 Alanis Morissette chillout
28075 0.521 Kanye West electronic
artist_neighbors(vanilla_model, "The Cure", COSINE)
Nearest neighbors of : The Cure.
cosine score name most assigned tag
9437 1.000 The Cure chillout
10850 0.981 Placebo chillout
4936 0.977 Depeche Mode chillout
16680 0.976 The Beatles chillout
8273 0.975 Radiohead chillout
43413 0.970 David Bowie chillout
artist_neighbors(reg_model, "The Cure", DOT)
Nearest neighbors of : The Cure.
dot score name most assigned tag
16680 3.297 The Beatles chillout
12363 3.217 Muse chillout
9437 3.213 The Cure chillout
18364 3.206 Nirvana pop
3259 3.186 Coldplay chillout
8273 3.177 Radiohead chillout
artist_neighbors(reg_model, "The Cure", COSINE)
Nearest neighbors of : The Cure.
cosine score name most assigned tag
9437 1.000 The Cure chillout
115196 0.970 Hole female vocalist
8273 0.970 Radiohead chillout
16680 0.964 The Beatles chillout
43413 0.964 David Bowie chillout
48889 0.959 R.E.M. atmospheric

We observe that dot product tends to recommends more popular artists such as Nirvana and The Beatles, where as Cosine Similarity recommends more obscure artists. This is likely due to the fact that the norm of the embedding in matrix factorization is often correlated with prevalence. The regularised model seems to output better reccomodations as the varation of the most assigned tag attribute is less when compared to the vanilla model. In addition, Marilyn Manson was recommended by the vanilla model in our intial run. We argue that these artists are most dis-similar! However, this observation is subject to change when you run the model, as we initialize the embedddings with a random gaussian generator.

def artist_embedding_norm(models):
  """Visualizes the norm and number of ratings of the artist embeddings.
  Args:
    model: A train_matrix_norm object.
  """
  if not isinstance(models, list):
    models = [models]
    df = pd.DataFrame({
          'name': artists.sort_values('artistID', ascending=True)['name'].values,
        'number of user-artist interactions': user_artists[['artistID','userID']].sort_values('artistID', ascending=True).groupby('artistID').count()['userID'].values,
    })
  charts = []
  brush = alt.selection_interval()
  for i, model in enumerate(models):
    norm_key = 'norm'+str(i)
    df[norm_key] = np.linalg.norm(model.embeddings["artistID"], axis=1)
    nearest = alt.selection(
        type='single', encodings=['x', 'y'], on='mouseover', nearest=True,
        empty='none')
    base = alt.Chart().mark_circle().encode(
        x='number of user-artist interactions',
        y=norm_key,
        color=alt.condition(brush, alt.value('#4c78a8'), alt.value('lightgray'))
    ).properties(
        selection=nearest).add_selection(brush)
    text = alt.Chart().mark_text(align='center', dx=5, dy=-5).encode(
        x='number of user-artist interactions', y=norm_key,
        text=alt.condition(nearest, 'name', alt.value('')))
    charts.append(alt.layer(base, text))
  return alt.hconcat(*charts, data=df)

artist_embedding_norm(reg_model)
def visualize_movie_embeddings(data, x, y):
  genre_filter = alt.selection_multi(fields=['top10TagValue'])
  genre_chart = alt.Chart().mark_bar().encode(
      x="count()",
      y=alt.Y('top10TagValue'),
      color=alt.condition(
          genre_filter,
          alt.Color("top10TagValue:N"),
          alt.value('lightgray'))
  ).properties(height=300, selection=genre_filter)
  nearest = alt.selection(
      type='single', encodings=['x', 'y'], on='mouseover', nearest=True,
      empty='none')
  base = alt.Chart().mark_circle().encode(
      x=x,
      y=y,
      color=alt.condition(genre_filter, "top10TagValue", alt.value("whitesmoke")),
  ).properties(
      width=600,
      height=600,
      selection=nearest)
  text = alt.Chart().mark_text(align='left', dx=5, dy=-5).encode(
      x=x,
      y=y,
      text=alt.condition(nearest, 'name', alt.value('')))
  return alt.hconcat(alt.layer(base, text), genre_chart, data=data)

def tsne_movie_embeddings(model):
  """Visualizes the movie embeddings, projected using t-SNE with Cosine measure.
  Args:
    model: A MFModel object.
  """
  tsne = sklearn.manifold.TSNE(
      n_components=2, perplexity=40, metric='cosine', early_exaggeration=10.0,
      init='pca', verbose=True, n_iter=400)

  print('Running t-SNE...')
  V_proj = tsne.fit_transform(model.embeddings["artistID"])
  artists.loc[:,'x'] = V_proj[:, 0]
  artists.loc[:,'y'] = V_proj[:, 1]
  return visualize_movie_embeddings(artists, 'x', 'y')

T-distributed stochastic neighbor embedding (t-SNE) is a dimensionality reduction algorithm useful for visualizing high dimensional data. We use this algorithim to visualise our embeddings of the regualrised model. Due to the large number of user submitted semantic categories, we decide to color-code the top 15 tags, with the rest being labelled as ‘N/A’. Although the sea of orange, indicating’N/A’, makes it difficult to interrupt these results, the regularised model seems to adequaltly cluster artists of a similar genre in it’s embeddings.

tsne_movie_embeddings(reg_model)
Running t-SNE...
[t-SNE] Computing 121 nearest neighbors...
[t-SNE] Indexed 17632 samples in 0.001s...
/opt/hostedtoolcache/Python/3.7.12/x64/lib/python3.7/site-packages/sklearn/manifold/_t_sne.py:793: FutureWarning: The default learning rate in TSNE will change from 200.0 to 'auto' in 1.2.
  FutureWarning,
/opt/hostedtoolcache/Python/3.7.12/x64/lib/python3.7/site-packages/sklearn/manifold/_t_sne.py:827: FutureWarning: 'square_distances' has been introduced in 0.24 to help phase out legacy squaring behavior. The 'legacy' setting will be removed in 1.1 (renaming of 0.26), and the default setting will be changed to True. In 1.3, 'square_distances' will be removed altogether, and distances will be squared by default. Set 'square_distances'=True to silence this warning.
  FutureWarning,
[t-SNE] Computed neighbors for 17632 samples in 4.843s...
[t-SNE] Computed conditional probabilities for sample 1000 / 17632
[t-SNE] Computed conditional probabilities for sample 2000 / 17632
[t-SNE] Computed conditional probabilities for sample 3000 / 17632
[t-SNE] Computed conditional probabilities for sample 4000 / 17632
[t-SNE] Computed conditional probabilities for sample 5000 / 17632
[t-SNE] Computed conditional probabilities for sample 6000 / 17632
[t-SNE] Computed conditional probabilities for sample 7000 / 17632
[t-SNE] Computed conditional probabilities for sample 8000 / 17632
[t-SNE] Computed conditional probabilities for sample 9000 / 17632
[t-SNE] Computed conditional probabilities for sample 10000 / 17632
[t-SNE] Computed conditional probabilities for sample 11000 / 17632
[t-SNE] Computed conditional probabilities for sample 12000 / 17632
[t-SNE] Computed conditional probabilities for sample 13000 / 17632
[t-SNE] Computed conditional probabilities for sample 14000 / 17632
[t-SNE] Computed conditional probabilities for sample 15000 / 17632
[t-SNE] Computed conditional probabilities for sample 16000 / 17632
[t-SNE] Computed conditional probabilities for sample 17000 / 17632
[t-SNE] Computed conditional probabilities for sample 17632 / 17632
[t-SNE] Mean sigma: 0.178678
/opt/hostedtoolcache/Python/3.7.12/x64/lib/python3.7/site-packages/sklearn/manifold/_t_sne.py:986: FutureWarning: The PCA initialization in TSNE will change to have the standard deviation of PC1 equal to 1e-4 in 1.2. This will ensure better convergence.
  FutureWarning,
[t-SNE] KL divergence after 250 iterations with early exaggeration: 77.263321
[t-SNE] KL divergence after 400 iterations: 2.773607
def m_embedding_norm(models):
  """Visualizes the norm and number of ratings of the movie embeddings.
  Args:
    model: A MFModel object.
  """
  if not isinstance(models, list):
    models = [models]
    df = pd.DataFrame({
          'title': artists.sort_values('artistID', ascending=True)['name'].values,
        'num_ratings': user_artists[['artistID','userID']].sort_values('artistID', ascending=True).groupby('artistID').count()['userID'].values,
    })
  charts = []
  brush = alt.selection_interval()
  for i, model in enumerate(models):
    norm_key = 'norm'+str(i)
    df[norm_key] = np.linalg.norm(model.embeddings["artistID"], axis=1)
    nearest = alt.selection(
        type='single', encodings=['x', 'y'], on='mouseover', nearest=True,
        empty='none')
    base = alt.Chart().mark_circle().encode(
        x='num_ratings',
        y=norm_key,
        color=alt.condition(brush, alt.value('#4c78a8'), alt.value('lightgray'))
    ).properties(
        selection=nearest).add_selection(brush)
    text = alt.Chart().mark_text(align='center', dx=5, dy=-5).encode(
        x='num_ratings', y=norm_key,
        text=alt.condition(nearest, 'title', alt.value('')))
    charts.append(alt.layer(base, text))
  return alt.hconcat(*charts, data=df)

Demo

You can find the most similar artist to a specified artist (that is contained in Last.FM) using the artist_neighbours() function. Similarily, you can find the top 10 recommendations of a particular userID [0 to 1891] using the user_recommendations() function. The first argument specifies the desired model, second argument the userID and third the top-k recommendations. Fourth argument represents the similarity measure, either DOT or COSINE (default = DOT, not a string).

user_recommendations(reg_model, 234, 10, COSINE)
score name most assigned tag
126582 0.939 Validuaté N/A
126513 0.919 Graforréia Xilarmônica rock
126400 0.903 The Vibrators punk
121134 0.901 7Seconds 80s
126554 0.899 Moreira da Silva N/A
126539 0.892 Menstruação Anarquika N/A
126490 0.891 Street Bulldogs N/A
126433 0.875 Violator thrash metal
126491 0.856 Bandas Gaúchas - www.DownsMtv.com N/A
126505 0.835 Planet Hemp rock

To further demonstrate the robustness of the system and measure the serendipity of our model, we incorporate the top artists that we listen to on Spotify (i.e. an unknown user). Note, these artists have to also be in the Last.FM dataset. The recommendation system should output similar artists based on it’s artist embeddings. The Spotipy library is used to interact with Spotify’s API. The similarity measure used is the Dot product. Due to the short lived nature of the spotify token and the fact you have to sign into a pop-up to retrieve the authentication token, we simply list our top 5 artists manually. If we did not, jupyter book will stall when attempting to build as it is waiting for our response. However, we provide the code used to retrieve the short-lived token for verification purposes.

"""
import spotipy
from spotipy.oauth2 import SpotifyOAuth
client_id = <insert_your_client_id>
client_secret = <insert your client secret>
redirect_url = '<insert your redirect uri>
scope = "user-top-read user-read-playback-state streaming ugc-image-upload playlist-modify-public"

authenticate_manager = spotipy.oauth2.SpotifyOAuth(client_id = client_id,client_secret = client_secret,redirect_uri =redirect_url,scope =scope,show_dialog = True)
sp = spotipy.Spotify(auth_manager=authenticate_manager)

artists_long = sp.current_user_top_artists(limit=5, time_range="long_term")
"""
top_5_artists =[
                 'Coldplay',
                 'Paramore',
                 'Arctic Monkeys',
                 'Lily Allen',
                 'Miley Cyrus'
]
spotify_reccomdations_df = pd.DataFrame()
for artist in top_5_artists:
  similar_artist_df = artist_neighbors(reg_model, artist)[['name','dot score']]
  spotify_reccomdations_df = pd.concat([spotify_reccomdations_df, similar_artist_df])
spotify_reccomdations_df.sort_values('dot score', ascending=False).head(10)
Nearest neighbors of : Coldplay.
[Found more than one matching artist. Other candidates: Jay-Z & Coldplay, Coldplay/U2]
Nearest neighbors of : Paramore.
[Found more than one matching artist. Other candidates: Paramore攀]
Nearest neighbors of : Arctic Monkeys.
[Found more than one matching artist. Other candidates: Arctic Monkeys vs The Killers]
Nearest neighbors of : Lily Allen.
Nearest neighbors of : Miley Cyrus.
[Found more than one matching artist. Other candidates: Miley Cyrus攀, Demi Lovato Ft. Miley Cyrus Ft. Selena Gomez Ft. Jonas Brothers, Miley Cyrus and Billy Ray Cyrus, Miley Cyrus and John Travolta, Hannah Montana and Miley Cyrus]
name dot score
3259 Coldplay 3.674
37842 Paramore 3.603
6543 Lady Gaga 3.585
12363 Muse 3.576
30355 Linkin Park 3.527
36290 Eminem 3.519
17472 The Killers 3.513
8965 Michael Jackson 3.506
30355 Linkin Park 3.505
17832 Green Day 3.501

We believe these recommodations are good as when our model was given an artist in the top five, it actually recommended other artits in the top five.

Evaluation Code

This is the code needed to produce the in-depth model comparison. As we decided to use different notebooks for different models, the results of this code will be combined and explained later in the book.

## create holdout test set for each user (15 items)
user_artists = pd.read_csv('data/user_artists.dat', sep='\t')
user_ids = []
holdout_artits = []
for user_id in user_artists.userID.unique():
  top_15_artists = user_artists[user_artists.userID == user_id].sort_values(by='weight').head(15).artistID.tolist()
  if len(top_15_artists) == 15:
    holdout_artits.append(top_15_artists)
    user_ids.append(user_id)
holdout_df = pd.DataFrame(data={'userID':user_ids,'holdout_artists':holdout_artits})

holdout_df.to_csv('data/evaluation/test-set.csv',index=False)
## Finding the models vanilla, regualrised predection for each user. 
def get_top_15_model_predictions(model, measure):
  """Computes the top 15 predictions for a given model
  Args:
    model: the name of the model
    measure: a string specifying the similarity measure to be used. Can be
      either DOT or COSINE.
  Returns:
    predicted_df a dataframe containing userIDs, their top 15 artists by the model, and the correspnding scores.
  """
  artist_name_id_dict = dict(zip(artists['name'], artists['artistID']))
  user_ids = []
  predicted_artists = []
  scores_list = []
  for new_user_id, orginal_user_id in orginal_user_ids.items():
    top_15_names = user_recommendations(model, new_user_id, k=15,measure=measure )['name'].values
    top_15_scores = user_recommendations(model, new_user_id, k=15, measure=measure )['score'].values.tolist()
    artist_ids = []
    for name in top_15_names:
      artist_ids.append(artist_name_id_dict[name])
    predicted_artists.append(artist_ids)
    user_ids.append(orginal_user_id)
    scores_list.append(top_15_scores)
  predicted_df = pd.DataFrame(data={'userID':user_ids,'predictions_artists':predicted_artists, 'score':scores_list })
  return predicted_df
# save the recommended artits into dfs and save them to data/evaluation folder
vanilla_dot_pred= get_top_15_model_predictions(vanilla_model, measure=DOT)
vanilla_cos_pred = get_top_15_model_predictions(vanilla_model, measure=COSINE)
reg_dot_pred= get_top_15_model_predictions(reg_model, measure=DOT)
reg_cos_pred = get_top_15_model_predictions(reg_model, measure=COSINE)

vanilla_dot_pred.to_csv('data/evaluation/vannila_dot_pred.csv',index=False)
vanilla_cos_pred.to_csv('data/evaluation/vanila_cos_pred.csv',index=False)
reg_dot_pred.to_csv('data/evaluation/reg_dot_pred.csv',index=False)
reg_cos_pred.to_csv('data/evaluation/reg_cos_pred.csv',index=False)